How Good Are You At Reasoning?

[Xei]

Here's some questions, some of which are quite cool, which you have to be quite mathematically minded to answer. I wanted to see how good people are at thinking rationally but creatively on DV. I threw in a couple of standard psychological experiments too. Just do the ones you can. Feel free to add more. 1. You have a row of 1,000 coins, all of which are heads up. You flip the second, fourth, sixth, and all other even coins over so that they're tails. Then you flip over the third, sixth, and all other coins which are multiples of three. Now you do this for every fourth coin, every fifth coin, etcetera, all the way up to every thousandth coin (which means just flipping the last one). The question is this: which coins are heads up, and why? There is a concise and surprising answer. 2. What's the maximum number of pieces you can cut a pizza into using a cheese wire (i.e. straight cuts) if you can only cut six times? The pizza is to thin to cut horizontally, and too hot to pick up and move around. 3. There's a 1% chance that the average person has cancer. Somebody with cancer has an 80% chance of testing positive when they go for a scan. Somebody without cancer has a 9.6% chance of testing positive (i.e. getting a false positive). You go for a scan and receive a positive result. What is the probability that you have cancer? 4. 100 people out of a group of 10,000 are tree-huggers. You ask all 10,000 people if they hug trees or not. 80 out of the 100 people who hug trees are honest about it and say yes. 950 out of 9,900 people who don't hug trees also pretend that they do. What fraction of people who claim to hug trees are genuine tree-huggers? 5. I have four cards with numbers on one side and colours on the other. I claim that cards with even numbers are red on the other side. I put the cards in front of you. They are 1, 2, red, blue. Which cards do you need to turn over to check my claim? 6. In the UK you can drink at 18. There are four people drinking at the bar. One is 21, one is 16, one is drinking coke, one is drinking beer. What do I have to check to make sure nobody is breaking the law? 7. In any group of (two or more) people, there are always at least two people with the same number of friends in that group. True or false? N.B. you can only be friends with somebody who is friends with you!

[Malac Reborn]

"How good are you at reasoning"? I think I'm OK, how about you?

[Xei]

Okay I hope. But only 15% of professional doctors get the one about cancer correct, so we're in the minority, aren't we? Assuming of course that you can answer it, which you showed no evidence of.

[Original Poster]

These are tricky. I'm guessing 1 instinctively for the first question and 14 slices for the second. For the 3rd, I'm a little puzzled. 72.32?

[DrunkenArse]

1, 5, and 6 are the ones that caught my eye. 1) If we let D(n) denote the number of divisors of n that are not 1, then the coins at positions n such that D(n) is even will be in their initial state. [edit] Better: The coins in positions with odd numbers of divisors will be heads up. [/edit] 5) To test the claim, one would turn over the card showing two and the blue card. When turning over the blue card, confirm that there's an odd number on it. (or there would be a blue and even card) When turning over the card showing two, confirm that it's red. 6) That nobody under 18 is drinking.

[Xei]

1, 2 and 7 are probably the most interesting to think about as puzzles. The first two have especially pretty answers. The middle four are more about human intuition and psychology. You are right about the odd number of divisors thing, but there is a more concise way of saying it.

[DrunkenArse]

Ok ok, squares will be in their original state. For 7, is friendship reflexive?

[Xei]

No, but it is symmetric. And of course, the point of the coins question is to provide a reason. Anybody can assert an answer after checking a few cases.

[LikesToTrip]

1. oxxoxxxxoxxxxxxoxxxxxxxx etc etc 2. 64 assuming I can reorientate the pizza between slices. 3. I'm a little confused on this one. I'm thinkin 1%. I'll ponder it a little longer though. 4. 80/1030 5. You didn't limit my card flips. I will flip them all. 6. The IDs of those drinking alcohol..... 7. False. You never claimed friendship had to be mutual. EDIT: looks like you changed the rules on the last question so my answer no longer applies. Don't have time to rethink it at the moment.

[Oneiro]

Okay, I'll bite: I'll probably fall into all the traps.. 1) I'm not atall mathematical, but I notice that the first and second coins remain untouched.. and seeing as you said that the answer was concise and surprising, I'm gonna guess 2; [BTW.. a "stack" of 1000 coins? How does one flip without dismantling the stack?] 2) Using 2D for a few minutes, I managed 19; but some sort of nagging voice is saying something about dual parallel perspective in 3D; 3) 4 in 5; 4) 80/1030ths = 40/515ths; 5) a) All of them: any card could have been tampered with; b) None of them: make sure that the cards are placed on a glass table and look from underneath; 6) First you need to check if the bar needs a license: it could be a bar built in one's living room, i.e. private- the UK 18-year-old thing only applies in certain premises; 7) False: could be any group of people who don't like each other.

[LikesToTrip]

7) False: could be any group of people who don't like each other. If they don't like each other. They all have 0 friends in the group so it would be true.

[Oneiro]

Oooh! Clever.. I said I'd fall into all the traps.. (Is zero a number?)

[Xei]

1. oxxoxxxxoxxxxxxoxxxxxxxx etc etc 2. 64 assuming I can reorientate the pizza between slices. 3. I'm a little confused on this one. I'm thinkin 1%. I'll ponder it a little longer though. 4. 80/1030 5. You didn't limit my card flips. I will flip them all. 6. The IDs of those drinking alcohol..... 7. False. You never claimed friendship had to be mutual. EDIT: looks like you changed the rules on the last question so my answer no longer applies. Don't have time to rethink it at the moment. 1. What do you mean by etcetera? 2. How come? 4. Yep. 5. But I did ask which ones you need to flip, and you don't need to flip them all. 6. Why would you need their IDs? The question tells you their ages. 7. Are you sure it'd be false? But okay, I clarified it. Okay, I'll bite: I'll probably fall into all the traps.. 1) I'm not atall mathematical, but I notice that the first and second coins remain untouched.. and seeing as you said that the answer was concise and surprising, I'm gonna guess 2; [BTW.. a "stack" of 1000 coins? How does one flip without dismantling the stack?] 2) Using 2D for a few minutes, I managed 19; but some sort of nagging voice is saying something about dual parallel perspective in 3D; 3) 4 in 5; 4) 80/1030ths = 40/515ths; 5) a) All of them: any card could have been tampered with; b) None of them: make sure that the cards are placed on a glass table and look from underneath; 6) First you need to check if the bar needs a license: it could be a bar built in one's living room, i.e. private- the UK 18-year-old thing only applies in certain premises; 7) False: could be any group of people who don't like each other. 1. I changed the question to a row of coins if that makes it clearer for you. 2. That is close to my answer. Don't worry about 3D, the pizza is 2D. 3. Way off. 4. Yep. 5. No tampering, no glass tables. None of these are trick questions. 6. See above. 7. What LikesToTrip said.

[Oneiro]

3) Oh yeah! I just read the question properly.. 1%;

[Xei]

It's definitely not 1% either.

[Oneiro]

I was referring to my brain power.. 2) I've managed 20 in 2D; A photon checks into a hotel and the concierge says: "Any luggage, Sir?" The photon replies: "No.. I'm travelling light.."

[Photolysis]

1. All primes will be tails. Since all non primes (save 1 which is not flipped) must have an even number of factors they will be flipped an even number of times and thus be heads. 2. I managed 21 playing around trying to get the maximum number of intersections with each cut. 3. 7.76% chance of actually having cancer with a positive result (or 800/10304 more precisely). For bonus points you have a 0.22% (or 200/89696 more accurately) chance of having cancer when you get a negative result. 4. 80/1030 obviously. Will look at the rest later.

[LikesToTrip]

1. What do you mean by etcetera? 2. How come? 4. Yep. 5. But I did ask which ones you need to flip, and you don't need to flip them all. 6. Why would you need their IDs? The question tells you their ages. 7. Are you sure it'd be false? But okay, I clarified it. 1. There will always be 2 more tails in between each heads all the way up to 1000. Why it does this, I have no clue. I'm sure there is some mathematical or logical equation you could write that sums it up, but that is beyond me. 2. Cut in in half and stack, cut in and half stack, cut in half and stack. You end up with 64 pieces. 2^6=64 5. Herp a derp. I'm retarded. Flip the 2. 6. You said four people. 21, 16, beer, coke. Aren't those the 4 people? We don't know the drinks of the 21/16 and we don't know the ages of the beer/coke. So we need to check the drink of the 16 and the age of the coke. Did I misinterpret the question? 7. I think the answer is now true. I'm not completely sure though. I'll ponder it more.

[Xei]

1. All primes will be tails. Since all non primes (save 1 which is not flipped) must have an even number of factors they will be flipped an even number of times and thus be heads. 2. I managed 21 playing around trying to get the maximum number of intersections with each cut. 3. 7.76% chance of actually having cancer with a positive result (or 800/10304 more precisely) 4. 80/1030 obviously. Will look at the rest later. 1. Along the right lines, but doesn't check out with the actual numbers (2 is non-prime but tails for example). 2. Pretty close. If you formalise your technique you should get it right. What about n cuts? 3. Yep; via what general method did you do this? Why was 4 a more obvious question? 1. There will always be 2 more tails in between each heads all the way up to 1000. Why it does this, I have no clue. I'm sure there is some mathematical or logical equation you could write that sums it up, but that is beyond me. 2. Cut in in half stack, cut in half stack, cut in half stack. You end up with 64 pieces. 5. Herp a derp. I'm retarded. Flip the 2. 6. You said four people. 21, 16, beer, coke. Aren't those the 4 people? We don't know the drinks of the 21/16 and we don't know the ages of the beer/coke. So we need to check the drink of the 16 and the age of the coke. Did I misinterpret the question? 7. I think the answer is now true. I'm not completely sure though. I'll ponder it more. 1. Try numbering the coins for a more concise description of which ones are flipped. The point of this question is in giving a good reason. I recommend you mull it over. None of these questions require mathematical knowledge or skill. They just require you to think like a mathematician, which is the important bit. 2. God damn you lateral thinkers. Okay no, no moving the pizza around. 5. Nope. You fell into the psychological trap that the vast majority of people do. 6. Yes that's completely right, and yes it is an easy question. The funny thing is this is logically identical to question 5.

[♥Mark]

You need to turn over 2 and blue. The claim is that all evens are red, however it's not specified that all reds are even. It doesn't matter if there is an uneven number on the other side of red. 2 is even and must be checked. Blue must be checked to ensure there is not an even number on it. 1 doesn't need to be checked since it isn't stated that odd numbered cards cannot be red.

[Malac Reborn]

I need to work on my math Original answer Okay I hope. But only 15% of professional doctors get the one about cancer correct, so we're in the minority, aren't we? Assuming of course that you can answer it, which you showed no evidence of. OK. 3) If I have cancer, then I have an 80% chance of being tested positive. If I'm tested positive, then the chance of me being positive is now 100%. The results, however, have only a 9.6% chance of being false, meaning I have only a 9.6% chance of not having cancer. Therefore, if I'm tested positive, then I have a 90.4% chance of having cancer.

[Photolysis]

1. Along the right lines, but doesn't check out with the actual numbers (2 is non-prime but tails for example). I was including 2 as a prime. Since when was it NOT a prime number, or am I misunderstanding you? 3. Yep; via what general method did you do this? Why was 4 a more obvious question? Easier numbers to work with. #3 also requires working out how many in total will get a positive result, but you already give the equivalent in #4. 2. Pretty close. If you formalise your technique you should get it right. What about n cuts? I'll have to think about a formal technique and on my answer.

[Oneiro]

5) Don't flip any.. I trust you, Xei;

[Xei]

2 is non-prime Hahaha what a fantastic retard I am. Okay, consider 6. Flipped at 2, 3, 6. Even number of factors means getting flipped to tails because 1 isn't included, that was my point.

[Malac Reborn]

Hey, Xei, could look at my answer and tell me where I went wrong?